13. Appendix

Computational engineering, especially linear algebra, has an abundance of material beyond what is reasonable to cover in this short and applied introduction. Included here are some material including explanations of useful algorithms, definitions, properties of certain matrices, and proofs that beg for some coverage.

Students are not accountable to know the material in the appendix for exams.

• 13.1. Norms
  • 13.1.1. Cardinality
  • 13.1.2. Vector Norms
  • 13.1.3. General Vector Norm, p-Norm
  • 13.1.4. Taxicab Norm, Manhattan Distance, or City Block Distance
  • 13.1.5. Euclidean Norm
  • 13.1.6. Infinity Norm
  • 13.1.7. vecnorm function
  • 13.1.8. Matrix Norms
  • 13.1.9. Maximum Absolute Column Sum
  • 13.1.10. 2-Norm of a Matrix
  • 13.1.11. Frobenius Norm
  • 13.1.12. Maximum Absolute Row Sum
  • 13.1.13. Nuclear Norm
• 13.2. Vector Spaces
• 13.3. Finding Orthogonal Basis Vectors
  • 13.3.1. Gram–Schmidt
  • 13.3.2. Implementation of Classic Gram–Schmidt
  • 13.3.3. Implementation of Modified Gram–Schmidt
• 13.4. Linearly Independent Eigenvectors Theorem
• 13.5. History of the Eigenvalue Problem
  • 13.5.1. Early Explorations
  • 13.5.2. Schur’s Insight
  • 13.5.3. Anticipation of Unitary Matrices
  • 13.5.4. Oak Ridge and Unitary Matrices
  • 13.5.5. The LR Algorithm
  • 13.5.6. John Francis
  • 13.5.7. Vera Kublanovskaya
• 13.6. All About the Number e
  • 13.6.1. Definition and Derivative
  • 13.6.2. Euler’s Complex Exponential Equation
  • 13.6.3. Numerical Verification of Euler’s Formula
  • 13.6.4. Compound Interest
• 13.7. A Matrix Exponent and Systems of ODEs
  • 13.7.1. A Matrix in the Exponent
  • 13.7.2. Matrix Solution to a System of ODEs
  • 13.7.3. Example ODE Matrix Solution